quote:

Originally posted by Psy-T optimistically speaking, now all that remains is actually writing it, and finding a publisher.

... and people that will read it.

oiy!

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Do any of you guys know anything about buying cars on ebay and having them shipped overseas? Because I'd really like to get my hands on a Dodge Charger, and as it seems they weren't sold in Europe so I'd have to get one from the USA. It's kind of a long-term plan that's not going to be realized sooner than in a couple of years' time because of a lack of finances, but still, it would be nice to know if someone has any experience on stuff like that.

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1+1=10

Lol @ Trancaholic: Numbers are tricky

quote:

Originally posted by tathi Where abouts in Brazil do you live Lira? I'll be in South America for about a year and hopefully Brazil for a month around New Years and Carnival, if you're in the area we'll ahve to catch up!

I live in Brasilia, which is a bit far from other major Brazilian cities (in spite of being the capital), but it's a nice place to visit because of the architecture. If you come here, do tell me

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quote:

Originally said by Adolf Fleisch Hitler How naive can people be? Curing the COR with messy hair and philosophy?

quote:

Originally said by Maurice Moss I came here to kick ass and drink milk... and I've just finished my milk

I posted this on another forum, but thought I might post it here for the edification of the maths wizards on this forum as well:

quote:

Okay, firstly these aren't homework questions: I've been reading "The Blind Watchmaker" recently and it's piqued my interest about the mechanics of evolution, so I'm really just asking these questions out of idle curiosity. Secondly, you shouldn't need to know much about evolution to answer these questions and the mathematics here shouldn't be too difficult, but I am absolutely useless with maths and wouldn't even know how to begin to set the equations up. Question 1: According to wikipedia, the odds of a person being spontaneously born with an extra digit are about 1 / 500. Presume that the gene for an extra digit is dominant (which is likely), so that a couple in which only one parent has an extra digit has about a 50% chance of producing a child with an extra digit and that the average couple has 2 children. Then presume that people born with an extra digit are 10% more likely to survive than those born without. In a population size of 1,000, how many generations will it take before the entire population has six digits? What if we presume a population size of 1,000,000 or 6 billion? Question 2: Presume that the mean length of the webbing on the hands of humans is 5mm. Presume the with each generation there is a natural 20% variance on the length of the webbing (i.e. the length of the webbing varies between 4mm and 6mm for the first generation) and that the data for each webbing length is evenly spread amongst the population. If we presume that each extra mm of webbing makes an individual 5% more likely to survive and reproduce, how many generations until the webbing of the entire population (again, let's say 1000, 1,000,000 and 6 billion) reaches all the way to the tops of the fingers (let's say, about 100mm)? Question 3: Similar to the last one, only assume it has to do with the height of animals. If the mean height of an animal is 100cm and there is a natural 5% variation either way (i.e. the range is 95cm - 105cm in the first generation) and that each additional cm means that the animal is 1% more likely to survive then how many generations will it be until the animal is 200cm tall? How about 1,000 cm? If there are any variables or assumptions I've missed in these scenarios, feel free to fill in the blanks yourselves or ask me about it. Thanks guys.

Anyone have any ideas?

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all the 999 members of this tribe are complete prudes, they will only reproduce with someone if that person has the same length of hand webbings as they do, they are completely monogamous, and they won't give birth to more than 2 offspring per couple, and to top that off, they're dying rapidly of a genetic disease to which the antibodies reside in the hand-webbings, some of the former members of the tribe who didn't have hand-webbings are all dead by now, they were estimated to have a 70% mortality rate, applicable before they had the chance to reproduce.. because that's when the disease attacks - before they can have an orgasm.
furthermore they discovered that each additional 1mm of hand-webbing lowers their mortality rates by 5%.
two more quirky things about this tribe are that their offspring always come out half with an added 1mm to their hand-webbings, and half with the same length as that of their parents, and the other thing, that comes somewhat incomprehensibly to my mind, is that currently their tribe is evenly divided to 3 groups - 1 with 4mm webbed-hands, 1 with 5mm webbed-hands, and 1 with 6mm webbed-hands.
the chief of the tribe sent me an email asking me for how many more generations will his tribe survive, if at all.

population 999
4mm 333
5mm 333
6mm 333

mortality
4mm 166
5mm 182
6mm 200
population 548

4mm produce 83 offspring with 5mm, and 83 with 4mm
5mm produce 91 offspring with 6mm, and 91 with 5mm
6mm produce 100 offspring with 7mm, and 100 with 6mm

4mm 83
5mm 174
6mm 191
7mm 100
population 548

mortality
4mm 41
5mm 95
6mm 125
7mm 65
population 326

4mm produce 20 with 5mm, 20 with 4mm
5mm produce 47 with 6mm, 47 with 5mm
6mm produce 62 with 7mm, 62 with 5mm
7mm produce 32 with 8mm, 32 with 7mm

4mm 20
5mm 67
6mm 109
7mm 94
8mm 32
population 322

mortality
4mm 10
5mm 37
6mm 66
7mm 62
8mm 22
population 197

4mm produce 5 with 5mm, 5 with 4mm
5mm produce 18 with 6mm, 18 with 5mm
6mm produce 33 with 7mm, 33 with 6mm
7mm produce 31 with 8mm, 31 with 7mm
8mm produce 11 with 9mm, 11 with 8mm

4mm 5
5mm 23
6mm 51
7mm 64
8mm 42
9mm 11
population 196

mortality
4mm 2
5mm 13
6mm 31
7mm 42
8mm 30
9mm 9
population 126

4mm produce 1 with 5mm, 1 with 4mm
5mm produce 6 with 6mm, 6 with 5mm
6mm produce 15 with 7mm, 15 with 6mm
7mm produce 21 with 8mm, 21 with 7mm
8mm produce 15 with 9mm, 15 with 8mm
9mm produce 4 with 10mm, 4 with 9mm

5mm 7
6mm 21
7mm 36
8mm 36
9mm 19
10mm 4
population 123

mortality
5mm 4
6mm 13
7mm 24
8mm 26
9mm 14
10mm 3
population 84

5mm produce 2 with 6mm, 2 with 5mm
6mm produce 6 with 7mm, 6 with 6mm
7mm produce 12 with 8mm, 12 with 7mm
8mm produce 13 with 9mm, 13 with 8mm
9mm produce 7 with 10mm, 7 with 9mm
10mm produce 1 with 11mm, 1 with 10mm

5mm 2
6mm 8
7mm 18
8mm 25
9mm 20
10mm 8
population 81

mortality
6mm 6
7mm 12
8mm 18
9mm 15
10mm 6
population 57

6mm produce 3 with 7mm, 3 with 6mm
7mm produce 6 with 8mm, 6 with 7mm
8mm produce 9 with 9mm, 9 with 8mm
9mm produce 7 with 10mm, 7 with 9mm
10mm produce 3 with 11mm, 3 with 10mm

6mm 3
7mm 9
8mm 15
9mm 16
10mm 10
11mm 3
population 56

mortality
6mm 2
7mm 6
8mm 11
9mm 12
10mm 8
11mm 2
population 42

6mm produce 1 with 7mm, 1 with 6mm
7mm produce 3 with 8mm, 3 with 7mm
8mm produce 5 with 9mm, 5 with 8mm
9mm produce 6 with 10mm, 6 with 9mm
10mm produce 4 with 11mm, 4 with 10mm
11mm produce 1 with 12mm, 1 with 11mm

7mm 4
8mm 8
9mm 11
10mm 10
11mm 5
population 38

mortality
7mm 3
8mm 6
9mm 7
10mm 8
11mm 4
population 28

7mm produce 1 with 8mm, 1 with 7mm
8mm produce 3 with 9mm, 3 with 8mm
9mm produce 3 with 10mm, 3 with 9mm
10mm produce 4 with 11mm, 4 with 10mm
11mm produce 2 with 12mm, 2 with 11mm

8mm 4
9mm 6
10mm 7
11mm 6
12mm 2
population 25

mortality
8mm 3
9mm 5
10mm 6
11mm 5
12mm 2
population 21

8mm produce 1 with 9mm, 1 with 8mm
9mm produce 2 with 10mm, 2 with 9mm
10mm produce 3 with 11mm, 3 with 10mm
11mm produce 2 with 12mm, 2 with 11mm
12mm produce 1 with 13mm, 1 with 12mm

9mm 3
10mm 5
11mm 5
12mm 3
population 16

mortality
9mm 2
10mm 4
11mm 4
12mm 3
population 13

9mm produce 1 with 10mm, 1 with 9mm
10mm produce 2 with 11mm, 2 with 10mm
11mm produce 2 with 12mm, 2 with 11mm
12mm produce 1 with 13mm, 1 with 12mm

10mm 3
11mm 4
12mm 3
population 10

mortality
10mm 2
11mm 3
12mm 3
population 8

10mm produce 1 with 11mm, 1 with 10mm
11mm produce 1 with 12mm, 1 with 11mm
12mm produce 1 with 13mm, 1 with 12mm

11mm 2
12mm 2
population 4

mortality
11mm 2
12mm 2
population 4

11mm produce 1 with 12mm, 1 with 11mm
12mm produce 1 with 13mm, 1 with 12mm

12mm 2
population 2

mortality
12mm 2
population 2

12mm produce 1 with 13mm, 1 with 12mm



as we can see here, despite the tribe's lowering mortality rates, the strictness of their rules led to their end within 14 generations, it's a sad tale.

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Last edited by Psy-T on May-26-2006 at 19:20

quote:

Originally posted by Renegade I posted this on another forum, but thought I might post it here for the edification of the maths wizards on this forum as well: Anyone have any ideas?

Ok, for the first guys, if they are 10% more likely to survive, I suppose their ratio in the general population grows by 10% each generation. Now, I suppose there's a one-line formula, but we can also go step by step.


1st generation - 2 6 fingered, 998 5 fingered
2nd - 2.200000 997.800000
3rd - 2.420000 997.580000
4th - 2.662000 997.338000
5th - 2.928200 997.071800
6th - 3.221020 996.778980
7th - 3.543122 996.456878

Hm..ok, this isn't really working..um..wait.

After n generations you have 2*(1.1)^n people with 6 fingers. So you wanna have a 1000 of them, then they'd exterminate the preceeding population. So, 2*(1.1)^n=1000 => 1.1^n=500 => n=log500/log1.1=65.8 generations. It's the same, regardless of the number of people (well, actually it isn't since with low populations you don't get whole numbers, but decimals - in that case every generation would have a 10% chance that one more 6 fingered kid will be born. So in that case the number of generations is partially based on luck)

The other two are pretty complicated to do. Like, for the webbings, you'd probably end up having all webbing dimensions, from 4-100mm, and you'd have to count each separately. Additionally, if people with mixed webbing lengths breed, what's their length? Median? I suppose you'd end up with some sort of a gaussian curve that would move a bit towards 100mm every generation..

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1+1=10

quote:

Originally posted by DrUg_Tit0 Ok, for the first guys, if they are 10% more likely to survive, I suppose their ratio in the general population grows by 10% each generation. Now, I suppose there's a one-line formula, but we can also go step by step. 1st generation - 2 6 fingered, 998 5 fingered 2nd - 2.200000 997.800000 3rd - 2.420000 997.580000 4th - 2.662000 997.338000 5th - 2.928200 997.071800 6th - 3.221020 996.778980 7th - 3.543122 996.456878 Hm..ok, this isn't really working..um..wait. After n generations you have 2*(1.1)^n people with 6 fingers. So you wanna have a 1000 of them, then they'd exterminate the preceeding population. So, 2*(1.1)^n=1000 => 1.1^n=500 => n=log500/log1.1=65.8 generations. It's the same, regardless of the number of people (well, actually it isn't since with low populations you don't get whole numbers, but decimals - in that case every generation would have a 10% chance that one more 6 fingered kid will be born. So in that case the number of generations is partially based on luck)

i'm not sure whether you worked with "a couple in which only one parent has an extra digit has about a 50% chance of producing a child with an extra digit and that the average couple has 2 children" as averages or imposed rules, because i don't understand the methods you used, but if it's the latter, i'm pretty confident the 6 fingered people can not rise signifinctly in number, but just in ratio to the 5 fingered people.

under the assumption that the statement quoted above is a rule, the only population rise for the 6-fingered people can come from "the odds of a person being spontaneously born with an extra digit are about 1 / 500".

quote:

Originally posted by DrUg_Tit0 The other two are pretty complicated to do. Like, for the webbings, you'd probably end up having all webbing dimensions, from 4-100mm, and you'd have to count each separately. Additionally, if people with mixed webbing lengths breed, what's their length? Median? I suppose you'd end up with some sort of a gaussian curve that would move a bit towards 100mm every generation..

what's wrong with the way i did that one other than the simplifications?

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quote:

Originally posted by Psy-T i'm not sure whether you worked with "a couple in which only one parent has an extra digit has about a 50% chance of producing a child with an extra digit and that the average couple has 2 children" as averages or imposed rules, because i don't understand the methods you used, but if it's the latter, i'm pretty confident the 6 fingered people can not rise signifinctly in number, but just in ratio to the 5 fingered people.

Hm, yeah, I really forgot to take notice of that part. Basically since the 5 fingered gene is recessive, it would result in a fact that there's going to be a much longer time until the 5 fingered gene disappears, since people can carry it and still have the 6-fingered advantage. Infact it may not disappear at all, ever. But that can't be calculated exactly, it's a matter of chance.

quote:

under the assumption that the statement quoted above is a rule, the only population rise for the 6-fingered people can come from "the odds of a person being spontaneously born with an extra digit are about 1 / 500".

Yes, that and that they have a 10% advantage.

quote:

what's wrong with the way i did that one other than the simplifications?

Well, nothing..except that humankind has not died out yet

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1+1=10

quote:

Originally posted by DrUg_Tit0 Well, nothing..except that humankind has not died out yet

cheeky!

the tribe died mostly because i set it to very closed circumstances where they had to reproduce only twice per cycle, and because the sample size i chose was the smallest, i'm pretty sure they would have survived; in a different situation, i calculated the same with a few changes, most importantly 3 reproductions per couple aswell as much harsher mortality rates - in that one, already on the 4th generation the fast dwindling of the tribe turned to a population increase.. on the 9th (or maybe it was the 10th) generation, the illness precentage in the tribe was around 6% and falling; by the 11th the tribe had a larger population than it did in the begining.

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quote:

Originally posted by Renegade I posted this on another forum, but thought I might post it here for the edification of the maths wizards on this forum as well: Anyone have any ideas?

I haven't had time to think much about this, but it appears to me that your first question cannot be answered in the absolute, as it builds on a stochastic variable I(X|Y,Z) ("Is individual X born with an extra digit given that its parents Y and Z have extra digits"), which is identically distributed for any two individual given the same configuration of their parents. (As (0.5, 0.5) for configurations where one parent has the extra digit, (0.002, 0.998) for configurations where no parent has the extra digit, and (what?) for configurations where both parents have the extra digit.) Therefore, all you can hope to achieve is a distribution over the relative number R_N of extra digit carrying members of the population after N generations, and not any absolute values. If you're willing to set a threshold value, you could get something like "how many generations until the probability of R exceeding 0.99 itself exceeds 0.95", or something similar, which might be what you're looking for.
Of course, such calculations would be horrible, and I doubt that there's a closed form expression for the density of R_N given the number of generations N. So if I were you, I would run a lot of simulations and then calculate the empirical means and variances of R_N for each generation.

Last edited by trancaholic on May-27-2006 at 10:11

Hmm, say anyone here have matlab and a lot of free time?

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1+1=10

Well, it's pretty much what I do for a living, and I guess I could mock up something non-fancy in Java in a couple of hours. It might even be fun. However, I would need to have the setup specified in much more detail:
- What is the probability of an offspring of two extra digit-individuals being an extra digit-individual itself?
- The reproduction selection mechanims is pretty unclear to me. If "the average couple" have two kids, then everyone will have to reproduce to keep the population growth from going negative (and hence answering the overall question trivially by having any such population go extinct). Additionally, it would mean that no natural selection is taking place this way.
Moreover, what does it mean for an individual to be "10% more likely to survive"? Does "survive" substitute for "procreate" here? And are individuals allowed to survive from one generation to the next (i.e. the same individual can compete for procreation "slots" in more than one generation)? Are individuals allowed to participate in more than one offspring generating couple in the same generation? And does "10% more" hint at "80% rather than 70% chance of surviving" or "77% rather than 70% chance of surviving".
Depending on how all these questions are answered the results may vary significantly.